The machine learning community has recently put effort into quantized or low-precision arithmetics to scale large models. This paper proposes performing probabilistic inference in the quantized, discrete parameter space created by these representations, effectively enabling us to learn a continuous distribution using discrete parameters. We consider both 2D densities and quantized neural networks, where we introduce a tractable learning approach using probabilistic circuits. This method offers a scalable solution to manage complex distributions and provides clear insights into model behavior. We validate our approach with various models, demonstrating inference efficiency without sacrificing accuracy. This work advances scalable, interpretable machine learning by utilizing discrete approximations for probabilistic computations.

While Large Language Models (LLMs) are often used as virtual tutors in computer science (CS) education, this approach can foster passive learning and over-reliance. This paper presents a novel pedagogical paradigm that inverts this model: students act as instructors who must teach an LLM to solve problems. To facilitate this, we developed strategies for designing questions with engineered knowledge gaps that only a student can bridge, and we introduce Socrates, a system for deploying this method with minimal overhead. We evaluated our approach in an undergraduate course and found that this active-learning method led to statistically significant improvements in student performance compared to historical cohorts. Our work demonstrates a practical, cost-effective framework for using LLMs to deepen student engagement and mastery.

A numeration system encodes abstract numeric quantities as concrete strings of written characters. The numeration systems used by modern scripts tend to be precise and unambiguous, but this was not so for the ancient and partially-deciphered proto-Elamite (PE) script, where written numerals can have up to four distinct readings depending on the system that is used to read them. We consider the task of disambiguating between these readings in order to determine the values of the numeric quantities recorded in this corpus. We algorithmically extract a list of possible readings for each PE numeral notation, and contribute two disambiguation techniques based on structural properties of the original documents and classifiers learned with the bootstrapping algorithm. We also contribute a test set for evaluating disambiguation techniques, as well as a novel approach to cautious rule selection for bootstrapped classifiers. Our analysis confirms existing intuitions about this script and reveals previously-unknown correlations between tablet content and numeral magnitude. This work is crucial to understanding and deciphering PE, as the corpus is heavily accounting-focused and contains many more numeric tokens than tokens of text.

Efficient number representation is essential for federated learning, natural language processing, and network measurement solutions. Due to timing, area, and power constraints, such applications use narrow bit-width (e.g., 8-bit) number systems. The widely used floating-point systems exhibit a trade-off between the counting range and accuracy. This paper introduces Floating-Floating-Point (F2P) - a floating point number that varies the partition between mantissa and exponent. Such flexibility leads to a large counting range combined with improved accuracy over a selected sub-range. Our evaluation demonstrates that moving to F2P from the state-of-the-art improves network measurement accuracy and federated learning.

In computation-intensive domains such as digital signal processing, encryption, and neural networks, the performance of arithmetic units, including adders and multipliers, is pivotal. Conventional numerical systems often fall short of meeting the efficiency requirements of these applications concerning area, time, and power consumption. Innovative approaches like residue number systems (RNS) and redundant number systems have been introduced to surmount this challenge, markedly elevating computational efficiency. This paper examines from multiple perspectives how the fusion of redundant number systems with RNS (termed R-RNS) can diminish latency and enhance circuit implementation, yielding substantial benefits in practical scenarios. We conduct a comparative analysis of four systems - RNS, redundant number system, Binary Number System (BNS), and Signed-Digit Redundant Residue Number System (SD-RNS)-and appraise SD-RNS through an advanced Deep Neural Network (DNN) utilizing the CIFAR-10 dataset. Our findings are encouraging, demonstrating that SD-RNS attains computational speedups of 1.27 times and 2.25 times over RNS and BNS, respectively, and reduces energy consumption by 60% compared to BNS during sequential addition and multiplication tasks.

This paper presents a mixed-computation neural network processing approach for edge applications that incorporates low-precision (low-width) Posit and low-precision fixed point (FixP) number systems. This mixed-computation approach employs 4-bit Posit (Posit4), which has higher precision around zero, for representing weights with high sensitivity, while it uses 4-bit FixP (FixP4) for representing other weights. A heuristic for analyzing the importance and the quantization error of the weights is presented to assign the proper number system to different weights. Additionally, a gradient approximation for Posit representation is introduced to improve the quality of weight updates in the backpropagation process. Due to the high energy consumption of the fully Posit-based computations, neural network operations are carried out in FixP or Posit/FixP. An efficient hardware implementation of a MAC operation with a first Posit operand and FixP for a second operand and accumulator is presented. The efficacy of the proposed low-precision mixed-computation approach is extensively assessed on vision and language models. The results show that, on average, the accuracy of the mixed-computation is about 1.5% higher than that of FixP with a cost of 0.19% energy overhead.

Deep neural networks (DNNs) have become an enabling component for a myriad of artificial intelligence applications. DNNs have shown sometimes superior performance, even compared to humans, in cases such as self-driving, health applications, etc. Because of their computational complexity, deploying DNNs in resource-constrained devices still faces many challenges related to computing complexity, energy efficiency, latency, and cost. To this end, several research directions are being pursued by both academia and industry to accelerate and efficiently implement DNNs. One important direction is determining the appropriate data representation for the massive amount of data involved in DNN processing. Using conventional number systems has been found to be sub-optimal for DNNs. Alternatively, a great body of research focuses on exploring suitable number systems. This article aims to provide a comprehensive survey and discussion about alternative number systems for more efficient representations of DNN data. Various number systems (conventional/unconventional) exploited for DNNs are discussed. The impact of these number systems on the performance and hardware design of DNNs is considered. In addition, this paper highlights the challenges associated with each number system and various solutions that are proposed for addressing them. The reader will be able to understand the importance of an efficient number system for DNN, learn about the widely used number systems for DNN, understand the trade-offs between various number systems, and consider various design aspects that affect the impact of number systems on DNN performance. In addition, the recent trends and related research opportunities will be highlighted

Representing deep neural networks (DNNs) in low-precision is a promising approach to enable efficient acceleration and memory reduction. Previous methods that train DNNs in low-precision typically keep a copy of weights in high-precision during the weight updates. Directly training with low-precision weights leads to accuracy degradation due to complex interactions between the low-precision number systems and the learning algorithms. To address this issue, we develop a co-designed low-precision training framework, termed LNS-Madam, in which we jointly design a logarithmic number system (LNS) and a multiplicative weight update algorithm (Madam). We prove that LNS-Madam results in low quantization error during weight updates, leading to stable performance even if the precision is limited. We further propose a hardware design of LNS-Madam that resolves practical challenges in implementing an efficient datapath for LNS computations. Our implementation effectively reduces energy overhead incurred by LNS-to-integer conversion and partial sum accumulation. Experimental results show that LNS-Madam achieves comparable accuracy to full-precision counterparts with only 8 bits on popular computer vision and natural language tasks. Compared to FP32 and FP8, LNS-Madam reduces the energy consumption by over 90% and 55%, respectively.

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It seems like there's a never-ending debate about autonomous vehicles and the neural networks that drive them. While opinions vary widely, many fall into the optimist and pessimist camps. The most optimistic of the optimists not only believe that building an autonomous vehicle is possible, but that an autonomous vehicle is conscious and alive in some way, or think that true artificial general intelligence isn't that far off. The pessimists think that not only will Tesla fail at creating Full Self Driving, but that all who try will fail. Even the pessimists who think Tesla and other companies may succeed often point to past failure or current safety concerns.